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Chapter 7: Q38E (page 346)

Consider a rotationT(x鈫�)=Ax鈫�in鈩�3in. (That is, A is an orthogonal 3x3matrix with determinant 1.) Show that T has a non-zero fixed point [i.e., a vectorT(x鈫�)=Ax鈫�in鈩�3withT(v鈫�)=v鈫�]. This result is known as Euler鈥檚 theorem, after the great Swiss mathematician Leonhard Euler (1707鈥�1783). Hint: Consider the characteristic polynomialrole="math" localid="1659595800447" fA(位). Pay attention to the intercepts with both axes. Use Theorem 7.1.4.

Short Answer

Expert verified

A is orthogonal, it applies .位=1.S0,鈭�x鈭�鈩�3,Ax=x,

Step by step solution

01

Theorem

The possible real eigenvalues of an orthogonal matrix are 1 and 鈭�1.

02

Solution of the problem

We have 0A=detA-0l=detA=1andim位鈫�鈭�,and , which means that 鈭�位鈭�0,鈭�,fA位=0. Since is orthogonal, it applies 位=1So,鈭�x鈭�鈩�3,Ax=x,

Therefore, the final result of is orthogonal, it applies So 位=1So,鈭�x鈭�鈩�3,Ax=x,

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Most popular questions from this chapter

In an unfortunate accident involving an Austrian truck, 100kgof a highly toxic substance are spilled into Lake Sils, in the Swiss Engadine Valley. The river Inn carries the pollutant down to Lake Silvaplana and later to Lake St. Moritz.


This sorry state, tweeks after the accident, can be described by the vector

x鈫�(t)=x1(t)x2(t)x3(t)pollutantinlakesilspollutantinlakesilvaplanapollutantinlakest.Mortiz}(inkg)

Suppose thatx鈫�(t+1)=[0.7000.106000208]x鈫�(t)

  1. Explain the significance of the entries of the transformation matrix in practical terms.
  2. Find closed formulas for the amount of pollutant in each of the three lakesweeks after the accident. Graph the three functions against time (on the same

axes). When does the pollution in Lake Silvaplana reach a maximum

Find an eigenbasis of given matrix and diagonalize it.

A=(1326)

Two interacting populations of coyotes and roadrunners can be modeled by the recursive equations

h(t + 1) = 4h(t)-2f(t)

f(t + 1) = h(t) + f(t).

For each of the initial populations given in parts (a) through (c), find closed formulas for h(t) and f(t).

(a)h(0)=f(0)=100(b)h(0)200,f(0)=100(c)h(0)600,f(0)=500

Question: If a vectorv鈫�is an eigenvector of both AandB, isv鈫� necessarily an eigenvector ofAB?

Find a 2脳2matrix A such that [31]and [12] are eigenvectors of A , with eigenvalues 5 and 10 , respectively.
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